Work out the information ratio on both an ex post and an ex ante basis, explain what it means, and contrast it with the Sharpe ratio

The information ratio divides active return by active risk. Ex post it is the mean realised active return over a period divided by the standard deviation of those active returns, the tracking error; ex ante it is expected active return divided by expected active risk. The Sharpe ratio instead divides return in excess of the risk-free rate by total volatility, so it rewards total risk-taking, not skill relative to a benchmark. Both ratios are unaffected by scaling: mixing an active fund with its benchmark scales active return and active risk together, just as mixing a portfolio with cash scales excess return and volatility. The trap is dividing active return by total volatility rather than by tracking error.